ISC Class 12: Mathematics Syllabus


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MATHEMATICS (860)

Aims: 
1. To enable candidates to acquire knowledge and to 
develop an understanding of the terms, concepts, 
symbols, definitions, principles, processes and 
formulae of Mathematics at the Senior Secondary 
stage. 
2. To develop the ability to apply the knowledge and 
understanding of Mathematics to unfamiliar 
situations or to new problems. 
3. To develop skills of - 
(a) computation. 
(b) reading tables, charts, graphs, etc. 
4. To develop an appreciation of the role of 
Mathematics in day-to-day life. 
5. To develop an interest in Mathematics. 
6. To develop a scientific attitude through the study 
of Mathematics. 
A knowledge of Arithmetic and Pure Geometry is 
assumed. The parts of Geometry which are of chief 
importance in other branches of Mathematics are the 
fundamentals concerning angles, parallels (including 
lines and planes in space), similar triangles (including the theorem of Pythagoras) the ‘symmetry’ properties of chords and tangents of a circle, and the theorem that a line perpendicular to two non-parallel lines in a plane is perpendicular to every line therein. The examination may include questions with a geometrical content. 
As regards the standard of algebraic manipulation, students should be taught: 
(i) To check every step before proceeding to the next particularly where minus signs are involved. 
(ii) To attack simplification piecemeal rather than en block, e.g. never to keep a common factor which can 
be cancelled. (iii) To observe and act on any special features of algebraic form that may be obviously present. 
The standard as regards (iii) is difficult to define; initial practice should be on the easiest cases, 'trick' examples should be avoided and it should be kept in mind that (iii) is subsidiary in importance to (i) and (ii) Teachers should be scrupulous in setting a standard of neatness and in avoiding the slovenly habit of omitting brackets or replacing them by dots. 

CLASS XI
There will be one paper of three hours duration of 
100 marks. The syllabus is divided into three sections
A, B and C. Section A is compulsory for all 
candidates. Candidates will have a choice of 
attempting questions from either Section B or 
Section C. 
Section A (80 marks) will consist of 9 questions. 
Candidates will be required to answer Question 
1(compulsory) and five out of the rest of the eight 
questions. 
Section B / C (20 marks) Candidates will be required 
to answer two questions out of three from either 
Section B or Section C. 


SECTION A 

1. Mathematical Reasoning 
Mathematically acceptable statements. Connecting words / phrases – consolidating the understanding of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”, “or”, “there exists” and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words, differences between contradiction, converse and contrapositive. 

2. Algebra 
(i) Complex Numbers 
• Complex numbers as an ordered pair of 
real numbers in the form a + ib, (a, b) 
• Geometrical representation in complex
plane - Argand diagram for z (a complex 
number), 1/z, z and z; equality of two 
complex numbers; absolute value 
(modulus). 
NOTE: Real and imaginary parts of a 
complex number and equality of two 
complex numbers are required to be 
covered. 
(ii) Quadratic Equations 
• Use of the formula 
in solving quadratic equations.
• Equations reducible to quadratic form. 
• Nature of roots 
− Product and sum of roots. 
− Roots are rational, irrational, equal, 
reciprocal, one square of the other. 
− Imaginary numbers. 
− Complex roots. 
− Framing quadratic equations with 
given root. 
NOTE: Questions on equations having 
common roots are to be covered. 
• Quadratic Functions. 
Given α, β as roots then find the 
equation 
Real roots, 
Complex roots, 
Equal roots 

- Sign of quadratic 
Sign when the roots are real and when they are complex.
- Quadratic inequalities.
Using method of intervals for solving problems of the type: 

(iii) Finite and Infinite Sequences 
(a) Arithmetic Progression (A.P.) 
(b) Geometric Progression (G.P.)
- Geometric Mean, b2 = ac
- Inserting 2 or 3 Geometric Mean between any two numbers. 
(c) Harmonic Progression (H.P.)
• a, b, c are in H.P then 1/a, 1/b, 1/c are in A.P. 

(d) Arithmetico Geometric Series 
• Identifying series as A.P. x G.P. (when we substitute d = 0 in the series, we get a G.P. and when we substitute r =1 the A.P.)
(e) Special sums 
∑n,∑n2 , ∑n3
• Using these summations to sum up other related expression. 


(iv) Permutations Combinations 
• Factorial notation n! , n! =n(n-1)! 
• Fundamental principle of counting. 
(a) Permutations 
• nPr
• Restricted permutation. 
• Certain things always occur together. 
• Certain things never occur. 
• Formation of numbers with digits. 
• Word building - repeated letters –No letters repeated. 
• Permutation of alike things. 
• Permutation of Repeated things. 
• Circular permutation – clockwise counterclockwise – Distinguishable  not distinguishable. 
(b) Combinations 
• nCr
 nCn=1, nC0 = 1, nCr = nCn–r, 
nCx = nCy, then x + y = n or x = y,
n+1Cr = nCr-1 + nCr
 . • When all things are different. 
• When all things are not different. 
• Division into groups - e.g. distinct groups, identical groups.
• Mixed problems on permutation and combinations. 

(v) Mathematical induction. Using induction to prove various summations and divisibility. NOTE: Problems on inequalities are not required. 

(vi) Binomial Theorem 
(a) Significance of Pascal’s triangle. 
(b) Binomial theorem (proof using induction) for positive integral powers, Simple direct questions based on the above. 
(c) Binomial theorem for negative or fractional indices 
When x <1
- Simple questions on the application 
of the above. 
- Finding the rth term for the above 
- Applying the theorem on approximations 
NOTE: Algebraic approximations are also  to be covered. 

(vii) Properties of Binomial Coefficients. 
Simple problems involving application of the above. 
NOTE: Questions on the product of coefficients of (1+x)n (x+1)m are excluded. 

3. Trigonometry
(i) Angles and Arc lengths 
• Angles: Convention of sign of angles. 
• Magnitude of an angle: Measures of Angles; Circular measure. 
• The relation S = rθ where θ is in radians. Relation between radians and degree. 
• Definition of trigonometric functions with the help of unit circle. 
• Truth of the identity sinx + cos2 x = 1. 
NOTE: Questions on the area of a sector of a circle are required to be covered. 
(ii) Trigonometric Functions 
• Relationship between trigonometric functions. 
• Proving simple identities. 
• Signs of trigonometric functions. 
• Domain and range of the trigonometric functions. 
• Trigonometric functions of all angles. 
• Periods of trigonometric functions. 
• Graphs of simple trigonometric functions (only sketches).
NOTE: Graphs of sin x, cos x, tan x, sec x, cosec x and cot x are to be included. 
(iii) Compound and multiple angles 
• Addition and subtraction formula: sin(A±B); cos(A±B); tan(A±B); tan(A + B + C) etc., Double angle, triple angle, half angle and one third angle formula as special cases. 
• Sum and differences as products 
• Product to sum or difference i.e. 2sinAcosB = sin(A + B) + sin(A – B) etc.

(iv) Trigonometric Equations 
• Solution of trigonometric equations (General solution and solution in the specified range). 
(a) Equations in which only one function of a single angle is involved e.g. sin 5θ =0 
(b) Equations expressible in terms of one trigonometric ratio of the unknown angle. 
(c) Equations involving multiple and sub- multiple angles. 
(d) Linear equations of the form acosθ + bsinθ = c

4. Calculus 
(i) Basic Concepts of Relations and Functions 
(a) Ordered pairs, sets of ordered pairs. 
(b) Cartesian Product (Cross) of two sets, cardinal number of a cross product. 
 Relations as: 
• an association between two sets. 
• a subset of a Cross Product. 
(c) Types of Relations: reflexive, symmetric, transitive and equivalence relation. 
(d) Binary Operation. 
(e) Domain, Range and Co-domain of a 
Relation. 
(f) Functions: 
• As special relations, concept of writing “y is a function of x” as y = f(x). 
• Types: one to one/ many to one, into/onto. 
• Domain and range of a function. 
• Composite function. 
• Inverse of a function. 
• Classification of functions. 
• Sketches of graphs of exponential function, logarithmic function, mod function, step function. 

(ii) Differential calculus 
 (a) Limits 
• Notion and meaning of limits. 
• Fundamental theorems on limits (statement only). 
• Limits of algebraic and trigonometric functions. 
NOTE: Indeterminate forms are to be introduced while calculating limits. 
(b) Continuity 
• Continuity of a function at a point x = a. 
• Continuity of a function in an interval. 
• Removable discontinuity.
(c) Differentiation 
• Meaning and geometrical interpretation of derivative. 
• Concept of continuity and differentiability of x , [x], etc. 
• Derivatives of simple algebraic and trigonometric functions and their formulae. 
• Differentiation using first principles. 
• Derivatives of sum/difference. 
• Derivatives of product of functions. 
• Derivatives of quotients of functions. 
• Derivatives of composite functions. 
NOTE: 
1. Derivatives of composite functions using chain rule. 
2. All the functions above should be either algebraic or trigonometric in nature. 
(d) Application of derivatives 
• Equation of Tangent and Normal approximation. 
• Rate measure. 
• Sign of derivative. 
• Monotonocity of a function.

(iii) Integral Calculus 
 Indefinite integral 
• Integration as the inverse of differentiation. 
• Anti-derivatives of polynomials and functions (ax +b)n, sinx, cosx, sec2 x, cosec2 x. 
• Integrals of the type sin2 x, sin3 x, etc.

5. Coordinate Geometry 
(i) Basic concepts of Points and their coordinates. 
(ii) The straight line 
• Slope and gradient of a line. 
• Angle between two lines. 
• Condition of perpendicularity and parallelism.
• Various forms of equation of lines. 
• Slope intercept form. 
• Two point slope form. 
• Intercept form. 
• Perpendicular /normal form. 
• General equation of a line. 
• Distance of a point from a line. 
• Distance between parallel lines. 
• Equation of lines bisecting the angle between two lines. 
• Definition of a locus. 
• Methods to find the equation of a 
locus. 
(iii) Circles 
Equations of a circle in: 
• Standard form. 
• Diameter form. 
• General form. 
• Parametric form. 
• Given the equation of a circle, to find the centre and the radius

Finding the equation of a circle
Given three non collinear points
Given other sufficient data that the centre is (h, k) and it lies on a line and two points on the circle are given. 
Tangents: Tangent to a circle when the slope of the tangent is given: 
Circle with a line hence to find the length of the chord.
Finding the equation of a circle through the intersection of two circles i.e. S1 + kS2 = 0
NOTE: Orthogonal circles are not required to be covered. 

6. Statistics
Measures of central tendency.
Standard deviation - by direct method, 
short cut method and step deviation 
method.
Combined mean and standard deviation.
Combined mean and standard deviation of  two groups only are required to be covered. Mean, Median and Mode of grouped and ungrouped data are required to be covered. 

Class 12 Syllabus


There will be one paper of three hours duration of 100 marks. The syllabus is divided into three sections A, B and C. Section A is compulsory for all candidates. Candidates will have a choice of attempting questions from either Section B or Section C. Section A (80 marks) will consist of 9 questions. Candidates will be required to answer Question 1(compulsory) and five out of the rest of the eight questions. Section B/C (20 marks) Candidates will be required to answer two questions out of three from either Section B or Section C. 

SECTION A 
1. Determinants and Matrices 
(i) Determinants 
• Order. 
• Minors. 
• Cofactors. 
• Expansion. 
• Properties of determinants. 
• Simple problems using properties of determinants
• Cramer's Rule 
ƒ Solving simultaneous equations in 2 or 3 variables, 
ƒ Consistency, inconsistency.Dependent or independent. 
NOTE: the consistency condition for three equations in two variables is required to be covered. 
(ii) Matrices 
• Types of matrices (m x n; m, n ≤3), order; Identity matrix, Diagonal matrix. 
• Symmetric, Skew symmetric. 
• Operation – addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices (the compatibility). 
• Singular and non-singular matrices. 
• Existence of two non-zero matrices 
whose product is a zero matrix. 
• Inverse (2x2, 3x3) 
• Martin’s Rule (i.e. using matrices) 
- Simple problems based on above.NOTE: The conditions for consistency of equations in two and three variables, using matrices, are to be covered
2. Boolean Algebra 
Boolean algebra as an algebraic structure, principle of duality, Boolean function. Switching circuits, application of Boolean algebra to switching circuits. 
3. Conics 
• As a section of a cone. 
• Definition of Foci, Directrix, Latus Rectum. 
• PS = ePL where P is a point on the conics, S is the focus, PL is the perpendicular distance of the point from the directrix.
 (i) Parabola 
• Rough sketch of the above. 
• The latus rectum; quadrants they lie in; coordinates of focus and vertex; and equations of directrix and the axis. 
• Finding equation of Parabola when Foci and directrix are given. 
• Simple and direct questions based on the above. 
(ii) Ellipse 
• Cases when a > b and a < b. 
• Rough sketch of the above. 
• Major axis, minor axis; latus rectum; coordinates of vertices, focus and centre; and equations of directrices and the axes.
• Finding equation of ellipse when focus and directrix are given. 
• Simple and direct questions based on the above.
• Focal property i.e. SP + SP′ = 2a.
(iii) Hyperbola 
• Cases when coefficient y2 is negative and coefficient of x2 is negative. 
• Rough sketch of the above. 
• Focal property i.e. SP - S’P = 2a. 
• Transverse and Conjugate axes; Latus rectum; coordinates of vertices, foci and centre; and equations of the directrices and the axes. 
• General second degree equation represents a parabola
Condition that y = mx + c is a tangent to the conics. 
4. Inverse Trigonometric Function 
• Principal values. 
• sin-1x, cos-1x, tan-1x etc. and their graphs. 
• Addition formulae. 
• Application of these formulae.
5. Calculus 
 (i) Differential Calculus 
• Revision of topics done in Class XI - mainly the differentiation of product of two functions, quotient rule, etc. 
• Derivatives of trigonometric functions. 
• Derivatives of exponential functions. 
• Derivatives of logarithmic functions. 
• Derivatives of inverse trigonometric functions - differentiation by means of substitution. 
• Derivatives of implicit functions and chain rule for composite functions. 103
• Derivatives of Parametric functions. 
• Differentiation of a function with respect to another function
• Logarithmic Differentiation 
• Successive differentiation up to 2nd order. 
• L'Hospital's theorem.
• Rolle's Mean Value Theorem - its geometrical interpretation. 
• Lagrange's Mean Value Theorem - its geometrical interpretation. 
• Maxima and minima.
(ii) Integral Calculus 
• Revision of formulae from Class XI. 
• Integration of 1/x, ex
• Integration by simple substitution. 
• Integrals of the type f' (x)[f (x)]n
• Integration of 1/x, ex, tanx, cotx, secx, cosecx. 
• Integration by parts. 
• Integration by means of substitution. 
• Integration using partial fractions, 
• Properties of definite integrals.
Problems based on properties of definite integrals are to be covered. 
• Application of definite integrals - area bounded by curves, lines and coordinate axes is required to be covered.
6. Correlation and Regression 
• Definition and meaning of correlation and 
regression coefficient. 
• Coefficient of Correlation by Karl Pearson. 
• Rank correlation by Spearman’s (Correction included). 
• Lines of regression of x on y and y on x. 
NOTE: Scatter diagrams and the following topics on regression are required. 
i) The method of least squares. 
ii) Lines of best fit. 
iii) Regression coefficient of x on y and y on x. 
v) Identification of regression equations 

7. Probability 
• Random experiments and their outcomes. 
• Events: sure events, impossible events, 
mutually exclusive events, independent 
events and dependent events. 
• Definition of probability of an event. 
• Laws of probability: addition and 
multiplication laws, conditional probability
(excluding Baye’s theorem). 
8. Complex Numbers 
• Argument and conjugate of complex numbers. 
• Sum, difference, product and quotient of two 
complex numbers additive and multiplicative 
inverse of a complex number. 
• Simple locus question on complex number; 
• Triangle inequality. 
• Square root of a complex number. 
• Demoivre’s theorem and its simple 
applications. 
• Cube roots of unity: ; application 
problems. 
2
1,ω,ω

9. Differential Equations 
• Differential equations, order and degree. 
• Solution of differential equations. 
• Variable separable. 
• Homogeneous equations and equations reducible to homogeneous form. 
• Linear form
NOTE: Equations reducible to variable separable type are included. The second order differential equations are excluded. 

SECTION B 
10. Vectors 
• Scalar (dot) product of vectors. 
• Cross product - its properties - area of a 
triangle, collinear vectors. 
• Scalar triple product - volume of a 
parallelopiped, co-planarity. 105
 Proof of Formulae (Using Vectors) 
• Sine rule. 
• Cosine rule 
• Projection formula 
• Area of a ∆ = ½absinC 
NOTE: Simple geometric applications of the 
above are required to be covered. 
11. Co-ordinate geometry in 3-Dimensions 
(i) Lines 
• Cartesian and vector equations of a line 
through one and two points. 
• Coplanar and skew lines. 
• Conditions for intersection of two lines. 
• Shortest distance between two lines.
NOTE: Symmetric and non-symmetric 
forms of lines are required to be covered.
(ii) Planes 
• Cartesian and vector equation of a 
plane. 
• Direction ratios of the normal to the 
plane. 
• One point form. 
• Normal form. 
• Intercept form. 
• Distance of a point from a plane. 
• Angle between two planes, a line and a 
plane. 
• Equation of a plane through the 
intersection of two planes i.e. - P1 + kP2 = 0. Simple questions based on the above.
12. Probability 
 Baye’s theorem; theoretical probability distribution, probability distribution function; binomial distribution – its mean and variance. 
 NOTE: Theoretical probability distribution is to be limited to binomial distribution only. 
SECTION C 
13. Discount 
 True discount; banker's discount; discounted value; present value; cash discount, bill of exchange. 
NOTE: Banker’s gain is required to be covered. 
14. Annuities
 Meaning, formulae for present value and amount; deferred annuity, applied problems on loans, sinking funds, scholarships. 
NOTE: Annuity due is required to be covered. 
15. Linear Programming 
Introduction, definition of related terminology such as constraints, objective function, optimization, isoprofit, isocost lines; advantages of linear programming; limitations of linear 
programming; application areas of linear programming; different types of linear programming (L.P.), problems, mathematical formulation of L.P problems, graphical method 
of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimum feasible solution. 
16. Application of derivatives in Commerce and Economics in the following 
 Cost function, average cost, marginal cost, 
revenue function and break even point. 
17. Index numbers and moving averages 
• Price index or price relative. 
• Simple aggregate method. 
• Weighted aggregate method. 
• Simple average of price relatives. 
• Weighted average of price relatives 
(cost of living index, consumer price index). 
NOTE: Under moving averages the following are required to be covered: 
• Meaning and purpose of the moving averages. 
• Calculation of moving averages with the 
given periodicity and plotting them on a 
graph.
• If the period is even, then the centered moving 
average is to be found out and plotted.